The mechanism of esterification (of anhydride + alcohol)

If the reaction were acid catalyzed, then it must be used in writing the mechanism. Hint, start with a protonation step by the acid. Zinc chloride is probably acting similarly. The actual species that must be protonated is not the actual most basic compound (ROH).

This mechanism is incorrect. It has been written as an uncatalyzed reaction (and chloride is not the most basic agent).

The zinc can coordinate with electrons in the same manner as a proton.

So how should look this mechanism properly?

Water is the by-product for a Fischer esterification, acetic acid is the by-product here. In your mechanism, you might want to "transfer" the zinc to an oxygen of the acetate mid-stage. Then it can be the weakest base. Why isnt the protonated alcohol the leaving group? Isn't it the weakest base?

Welcome to the forum.  You may wish to read over the forum rules, so that you can better understand what it is that we do and do not do here.  It is up to you to write the mechanism out, and then we can give you pointers.

I do not know how the end of the mechanism will look like. According to the instructions I have water as a product.